I NUC TP 544

T,)Op,

iI

I I' .

00 FRICTION COEFFICIENTSI •OF• •SYNTHETIC ROPES

byW. E. Brown

SO C E A N T E C H N O LO G Y D E P A R T M EN T- February 1977SI

R C'

1 Approved for public release, distribution unlimited

"g: •

BestAvaillable

Copy

R. 8. GII.CNRIST, CAP", USN HOWARD L. BLOOD, PhDCommander Technical Director

ADMINISTRATIVE INFORMATION

The work in this, report was sponsored by the Naval Sea Systems Command (code0112) under program element 60000N. Work was performed from June through September

Part of the work reported in this document was performeq bi General TechnologyCompany under contracts N6600l-76-4-Ai 3 and N66001-76-M-A$ý 14 under the direction

of Dr. Taghi J1. Mirsepassi.

Released by .Under authority ofA. J. SCHLOSSER, Head H. R. TALKINGTON, HeadApplied Technology Division Ocean Technology Department,4

............ .................... ...

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SUMMARYL;

PROBLEM

LJ 1. Develop a measurement technique and determine the friction coefficients betweensynthetic ropes and capstans and synthetic ropes and bitts.

1 i2. Develop a measurement technique and measure the effects of surface heating on- various rope diameters and load tensions.

3. Evaluate resulting data for applications in marine technology.

RESULTS

1. Test data for consecutive runs on the same rope-capstan combination show a con-"siderable range of variation.

2. More than half of the rope-capstan combinations tested indicated higher frictioncoefficients for a wet condition.

3. The chafing effect of capstans of stainless steel and NiCr/Cr 3C2 plasmalloy surfacescan seriously affect the life of synthetic ropes.

4. No chattering or vibration of the capstan-rope combinations was observed duringthe tests.

5. There is a large difference in the friction coefficient for bitts and capstans becauseof their different surfaces.

6. The results of the bitt tests agree with those of the low-load capstan tests.

RECOMMENDATIONS

1. Use the test apparatus and data evaluation method to compare the effectivenessof various coatings and treatment procedures for improving the rendering qualities of syn-thetic ropes.

2. Further investigate the effects of a wet rope-capstan combination, i.e., its higherU friction coefficient, on marine applications.

3. Investigate the effects of temperature and surface melting on the friction

coefficient.

LAII

Ut

CONTENTS

INTRODUCTION 3

MATHEMATICAL DERIVATIONS 3

Friction Coefficients 3

Friction Heating of Line 9 1CAPSTAN TESTS 11

Test Apparatus 11

Coefficient Detetmination 17

Measurement Errors 19

Test Conditions 22

Test Results 29

BITT TESTS 36

Apparatus and Procedures 36

Test Data and Computed Friction Coefficients 38

APPLICATIONS 40

Introduction 40

Stresses in Bitts and Bollards 41

Rope Elongation Effect 49

Hoisting, Mooring, and Towing 54

DISCUSSION 65

SREFERENCES 66

2

INTRODUCTIONSince the introduction of man-#ade fibers, e.g., nylon, dacron, and polypropylene,

• synthetic fiber ropes have becohiýavailable for use aboard ships. These ropes are superior tonatural fiber ropes in strength, wear, and resistance to rot, decay, and marine fungus (refer-,ence 1). However, the inethods of hand~hng natural fiber ropes do not directly apply to thesynthetic ropes, and disregard for the differ'ent physical and mechanical properties can cause

Sdamage and occasionally ftlacdns

The rendering' 4uality of the. synthetic ro'pes, i.e., their ability to move smoothlyove bitts and ca-4p-giaxis without siknorcatigis of fundamental interest and iprtance in marine •ipplicatiOnis. A good knowledge of the predominant factors, e.g., frictionand surface meltfing, is dssential in developing-ef~fective handi~ng techniques and" preventing

i• fatal accidents, excessive surface melting, and premature failure. It is the purpose of this

report to acquire such data by determining the static and sliding friction coefficients of syn-thetic ropes on bitts and capstans and the effects of" surface heating on different rope

S~diameters.

Friction between dry surfaces is the result of the microscopic and submicroscopicr• roughness of the two surfaces that come in contact. At rest, the interlocking of the protu-i;•_jberances of the contacting surfaces necessitates application of a higher tangential force T to

cause sliding (figure 1). However, once sliding starts, some protuberances are sheared off-• and the moving body rides over the bumps of the fixcd body; consequently, a lower force is

needed to maintain the motion.

J9

S~The tangential force required to begin the motion is called the "static friction force"or te"impending sliding frcinforce"; tefrerequired to keep tebody sliding isth

"sliding friction force." Both vary with the magnitude of the normal force N, and within alimited range they are linearly dependent on the intensity on N.!•

The friction coefficients are defined as ratios of the friction forces to the normal!" 'force N: --'

F.coefficient of static friction =yg=-f

I]I

coefficient of sliding friction = N --- ,:

13

Smoving body N direction Of motion 0

frictionforce

%I

fixed body

Figure 1. Static and sliding friction forces.

where Fu and F' denote, respectively, the ultimate static and the ultimate sliding frictionforces. The basic equations are

Fu

and

(2)

where the unprimed symbols refer to the impending slide condition and primed symbolsrefer to the sliding condition.

For synthetic lines wrapped around capstans or bitts, factors which affect- the fric-tion coefficients A and t' are rope and capstan materials, surface roughness of the capstan,abrasion of ihe rope and its surface texture, and the load tension on the line. Furthermore,when sliding occurs, the temperature of the rope,, its surface softening and melting charac-teristics, and the internal friction forces of the rope affect the friction coefficients.

To derive the pertinent equations, a rope wrapped around a fixed cylindrical post isconsidered (figure 2). The tensions applied to the rope are T 1 and T2 (T1 > T2). The con-tact angle of the rope with the cylinder is ot. The differential equation relating the tangentialforce T around the contact region of the rope with the variable angle 0 is obtained by con-sidering an infinitesimal section dO of the rope and writing the equilibrium equation.

4 "

M - - - ~ - -

4~d

T,11%

IIT2

dO

~. 2

NI 3RNii /

I I

Figure 2. Derivation of the basic differen~tial equations.

The frte&b6dy diragtfi~iii figure 2 is for a section of the food, dkxteding& from 0 to."0 + dO. Fbr this section, the equations of static equilibrium at the moment of iMpendinig-slide Are as followg:

Sforcý4componenf's A16ngtangtnfit40-.

Therefore ~ frecm6esl~g6~a 0

(T + fit) cos d5 )- cos (dO/2ý) -dFu 0- (3)

dN - (T +dT) sin (dO/2-)-- T sin -(1/2) 0. ()

Letting dO z0: (d/)±sO 1

sin (dO/2) "~d'/2.

Noting that dT (d6/2) is n~gligible compared to T(dO/2), equations 3 a~nd 4 redluce to x

dT -dF =()

4 j dN - T dO 0.(6)

'I he ultimate static friction force dFullof the incremental section dO can be expressedIn terms of dN by means of equation 1:

dFu!LdN.

Equation 5 becomes

dT - pi dN 0'. (7)

Elimination of dN between equatiofis-6 and- 7 -rtsultg i'n-.

dTju = d 0 (8)

which is the basic differential equation relating T(O), 0, and A.

The general solution of equation 8 is of the form:

T Cexp(IAO) ,()

6

RRM 77 ''F

[41

where C is the constant of integration and can be found from the end condition:

T T2 at 0=0

Thus C is foiuhd by

LI ~C exo(0) C

and equation 9 becomes -UJ

T(0)*T 2 expOi(o); (10)

Equation 160 witte•d6 0 = - results in

T T2 6xp(picO. (

" im pEn uati~ n l 1 p6i•idd the ielatiohshijý amdiig Ti,T-2, , aid 1A at the conditionf of-[ impending slide. T'hus, if ot, T 1, andT12 are known, p is found from the expression:

- In (T1I/T2)

(12)

where In (T1 /T 2) denotes the logarithm in base e of Tl/T 2. As defined earlier, the fiic-SfU tion coefficient A found in this manner is the static friction coefficient.

Example: Ship Docking

SBAs an application of equation 1, the problem of docking a ship is presented. The ob-jective is to slow down, and gradually stop a ship moving along a dock by means of a linesecured to-the dock at a point aft of the ship. -Control of the force applied to the ship isaccomplished by the pull of a seaman at the other end of the line on the deck. By wrappingthe rope on the deck of the ship around two bitts in a figure-eight fashion, the pull requiredfrom the seaman is quite small compared to the pull required for stopping the ship (figure 3).

The force to be applied to the ship is assumed 80,000 pounds, and the bitts, ,lfoot-diameter posts, are 2 feet apart. The problem is to find the required number of figure-eight

171 wrappings, assuming that the seaman cannot ipply more than a 60,pound pull

Assuming a.slide friction coefficient

L=0.4 ,equation 411 results in a required contact angle of

80,000 = 60 exp(O.4 a). Uk

7

, -, .~

D:

T2 >60 1b

0.5 t"

60dg240Odeq l240 deg - /.-

S~~ship,

T1 80,000 Ib

dock*

Figure 3. Ship docking problem;

Therefore;

I ln (80,000/60) _ 18 radians. jk0.4If n denotes the number of figure-eight wraps, then the total line contact angle a will

be L

ot = n (2 X 240 degrees) + 240 degrees = n (2 X 4.189) + ,4.189 radians.

Based on this equation and the minimum required contact angle of 18 radians, n is found:

n = (18 " 4;189)/(2 X 4.189) = 1.64.

Therefore a total of two figure-eight wraps around the two bitts is required.

>; ~-4 Example: Load Pulling Using a Capstana

To show another application, the problem of pulling a load by, means of a capstan is,PEI +discussed. The problem is to find the number of line wrappings around a motor-driven

8

capstan for pulling a load of 40,000 pounds, assuming that the anplied tension at the relaxed

end of the line is 40 pounds. In this examiple

T1 =T40,000 pounds

T2 40 pounds

and for an assumed slid6 friction coefficient

Equation 11 becomes

40,000 =40 exp(0.4 c4),

from which ca is found:

Of in (40,600/40) 17.25 radians.L_ 0.4

Therefore, a minimum number of wrapping

"• ~17.25na5 23 2.75 = 3 turnsi2 X .14

is required (figure 4).

FRICTION HEATING OF LINE

When a line is wrapped around the bitts to apply a controlled tension to the line (as

I ~j in the first example), the line is allowed to slide in order to limit the maximum tension. In

this case the friction force (T1 - T2 ) translates along itself and performs mechanical work-J

which- is transformed into heat energy. Thus, if at an instant in time, the line is slidingaround the bitts at the rate of v, feet per second, the rate of heat generatior along the con-

tact length, Rc of the contact area per unit length of rope is given by

v(TI -T2 )0 btu per secondq c XO0 feet of rope

Each small section AR of the rope in its sliding motion will be in contact with the bitts for

A At - seconds. (14)

,gV

AUý

-- 4 .- -4, ,-- 49

T2 =40 Ib 4 Z Z __

T= 40,000 Ib

Figure 4. Use of capstan to pull a load.

During the time period At, the rope-bitt interface will be subject to a hea.:U.Ig rate ofv (T1 - T2)

q A = 1 - ) X 0.001285 AQ btu per second. (15)

To •'nd the temperature rise of the rope surface, it is necessary to determine therope-bitt interface's temperature at the end of the At time period. Since At is short, the ropeand bitt may be considered as semiinfinite bodies, i.e., flat and large enough so that the heat-ing effect does not penetrate too far from the interface. With this simplifying assumption,an analytical expression can be derived to predict the rope's surface temperature. However,there are other factors, such as the rope's thermal conductivity as affected by the rope's te,-x-ture and structural nonhomogeneity and the occurrence of rope melting and the ensuingthermal, structural, and frictional effects, that make the analytic approach hopelessly in-volved and consequently unwarranted. For this reason, it would be advantageous to deter-mine the friction heating effects by instrumentation and direct measurement of the rope'ssurface temperature under various load tensions. (This area will be investigated at a laterdate.)

'•:• 10

.•7.,.•."*..,•c . . • •.. • . .• • ¢j.,; •.• • •,•.•• •, •. • • • .: • k•" _• :,:' ' I•

'.. . . .. . . . .77- 7 - -

CAPSTAN TESTS

TEST APPARATUS

Inclined Plane Method

The standard method of finding the friction between two surfaces is the inclined-- plane method (figure 5), where the inclination of the plane is gradually increased until sliding

starts. The friction coefficient is then given by 4

p p=tan 8, (16)

1 where 8 is the inclination angle.

'1 There are, however, numerous considerations regarding the friction of syntheticropes on bitts and capstans which make this approach difficult to implement. For example,

] the tension load on synthetic ropes affects the stiffness of individual fibers and consequentlythe rendering qualities and surface abrasion property of the rope. Therefore, in any test set-" ; up provision should be made for varying the rope's tension over the range Of practicalinterest. Furthermore, in an actual application, although both ends of the line are under

"tension, there is a large difference between the magnitude of the two. This condition mustbe duplicated in the test apparatus, if meaningful data are to be obtained. Other problemsinclude (1) the cylindrical shape of the bitt surface; (2) the heating effect of the frictionwhich accumulates as the rope slides over the bitt and eventually causes the surface strandsof the rope to meet; and (3) the large number of combinations of parameters: load tension,rope dimension, and rope texture.

jr {>. 60 deg

440 degTT

NNI°°1-11

8 Figure 5. Standard method of measuring friction coefficient.

• ,!II• q I l n l I I11

RI

Capstan Method

With these problems in mind. the test set up described in this section was developed.

The test apparatus (figures 6 through 9) basically consists of two capstans installedwith their axes horizontal. The test capstan shares a shaft with a larger diameter sheave andcan thus rotate freely. The fixed capstan cannot rotate, but is installed on a platform which1,

slides on rollers towards or away from the test capstan. The two are of equal diameter, andthe test capstan can be interchanged with capstans of various surface roughnesses.

The ends of the ropes to be tested are spliced to form a continuous loop which isthen wrapped several turns around the fixed capstan and passed once (half-turn) around thetest capstan. Tension is applied to the loop by applying a traction force to the sliding plat-form of the fixed capstan. The required tension is developed by filling barrel B I with water.Tension gauge G 1 measures the weight of the barrel and its content. (The weights of thegauge and the cable and the friction of moving parts are used in the tension calculations.)

The counter static friction force, which eventually overcomes the friction force andcauses sliding, is introduced by gradually filling the barrel B2 with water. The weight of thehanging barrel and its water is indicated by means of tension gauge G2. (The weight of thevertical segments of the cable and the ratio of the sheave groove's effective diameter to thediameter of capstan are used in calculations of the friction force.)

steel cable test capstan=• sheave fixed capstan

B2

GI M

steelcable B1 i

' Bi

Figure 6. Test apparatus.

12-A-... .• • "' _ • "_ . . ' ••• _ "a i. . •. . i ~ • h •' "•i

4 '4~-'

~ ~f- -4P

II

Part A. Westh-northwest view. 1

Fiur 7.Tsie-p

~~13

.2

" ! ai

• \I

•-A

Part C. East-southeast view. (Tension gauge G2 is missing since it was

replaced with a platform scale for greater accuracy.)

•:'•. Figure 7. Continued.1

14-

�-'. - __________

r - ---- �-----�-------- -

-�

AU I

a.. F

7. 7

P

Ii

�1

a (

$

Figure 8. Barrel B2 and platform scale used to determine its weight.

-� i 15

� -.- �,. P -- p

t4 uP 1v

al1 .

I Uý

tZtr A

2 ii-/ - AK'

RE

4 L,

Fiur 9. BarlBIadteso ag

164 I 3

II ! ! ! ! J I, ,T

SThe effective perimeter of the sheave groove for a 3/84inch steel cable is measured at

9 feet 2 inches. From this perimeter the effective diameter D is found:

9 ft 2 inD= 35 in.

iJ The perimeter of the capstan in the middle cylindrical section is measured at 47-3/4 inches,k Iwhich results in a computed diameter d:

d= 47-3/4 inid= ir =15.2 in.

; • COEFFICIENT DETERMINATION

The expressions relating the rope tensions T1I and T2, the wrap angle ci, and the

coefficient of friction L are as follows:

W=T 1 +T 2 (17)

{](TI -T2) 4=Wc7 (18)

.• #=-In (TI/T2), (TI >T2) (19)

Based on these equations, the following expression is derived for the coefficient of

"friction p as a function of W and Wc:

! •W +Wc (D/d)

pI=(l/ot) nw_Wc (Did) (20)

Ell (See figure 10.)

fi In the above expressions, the frictions of the pulleys and sliding platform have beendisregarded. If the force applied at the point of action of Wc (A in figure 10) to overcomethe rotational friction of the pulley, steel cable, and sheave-capstan assembly is denoted by

SF and if the force applied at the point of action ofW (B in figure 10) to overcome the fric-tion of the sliding platform, steel cable, and pulley is denoted by Ft, equation 20 is modifiedas follows:

p•l/a) (W - Ft) + (Wc - Fc) (D/d)1/a n -Ft)In Fc) (D/d) (21)

The above formula can be used to compute the friction coefficient from the test data.

17= =

Wc sheave dT-t- sta fixed capstan

A BB2 B,•• ~ ~D = sheave diameter •I

d = capstan diameterW = tension force (weight of 11, G1, and cable)

c W Wc = counter friction force (weight of B2, G2, and cable)S= wrap angle (it radians)

w

Figure 10. Definition of symbols.

The static friction coefficient is obtained by filling barrel B 1 to the level where a de-sired tension load W is reached and then gradually filling barrel B2 until the test capstanstarts to slide against the rope and allows the sheave-capstan assembly to rotate. The corn-bined weights of the barrel, its contained water, and the vertical segments of the cable areWc in equation 21.

The friction force Fc is found by determining the weight which must be applied to3 i point A of the cable to cause the sheave-capstan assembly to rotate in the absence of ropeS~friction. Likewise, Ft is found by determining the force which must be applied to the cable

on the tension side to cause motion of the sliding platform in the absence of rope tension.

In the above sequence of operations, once the static friction force of the rope on thetest capstan is overcome and sliding has started, the coefficient of friction between the ropeand the capstan decreases, resulting in an accelerated fall of barrel B2. To find the slidingfriction, the weight Wc must be reduced until a small initial velocity applied to the cablecauses a uniform constant-speed fall of the barrel. If the weight Wc is less than the Wc,sl re-quired for a uniform fall, then, despite the initial sliding velocity, barrel B2 will stop in ashort distance; if Wc is greater than Wc,sI, the initial sliding velocity results in an acceleratedfall of the barrel. Thus, the exact value of Wc,sl can be found by increasing the weight We,i.e., the amount of water in B2, by small increments until such time that a small initial slid-ing velocity applied to B2 results in its continuous fall.

18

With Wcsl determined in this manner, the sliding coefficient of friction /sl is foundfrom equation 21 by replacing Wc with Wcsl, i.e.,

(W - Fc) + (Wc,sl - Fc) (D/d)jusl = (IlIc) • In (W - Fc) - (Wc,sl - Fc) (D/d) "(22)

Since Wc,sl is less than the Wc required for overcoming the static friction force, the counter

friction force of sliding Wc,sl is found and then the counter friction force of static conditionWc. In this manner, the water is constantly added to B2 and there is no need for emptyingthe barrel between the two tests.

MEASUREMENT ERRORS

i " •The measurement errors are from two sources:

Errors in measuring tension and counter friction forces, W and Wc (or Wcsl),respectively

SErrors resulting from uncertainties in the static and dynamic frictions associated withthe sliding platform rollers, steel cables, pulleys, and bearings

To reduce errors from the tension gauge indications, both were calibrated (tables Iand 2). As time allowed, many tests were performed with tension gauge G2 removed and theweight of barrel B2 measured directly on a platform scale. This procedure, despite a con-siderable increase in test time, provided highly accurate results. The inaccuracy in readingthe platform scale is about 1/4 pound, which for a low total barrel weight of 50 poundsmeans a negligible error of

0.5"-0 0.005 or 0.5 percent,

which can be compared to an error of I to 5 percent for the tension gauge.

The errors resulting from uncertainties in the static and dynamic frictions of themoving parts of the apparatus, e.g., sliding platform and pulleys, enter into the calculationsdirectly and must be estimated as accurately as possible. For this purpose, static tests of thesliding and the sheave-capstan assembly were conducted. Results showed the followingvalues for dry friction:

_ 15 pounds for the sliding platform rollers and associated cable pulley• 8 pounds for the sheave-capstan assembly and associated cable and pulley

Once motion starts, the contact frictions are reduced, but the cable's internal friction isadded. Since an accurate measurement of the dynamic frictions requires elaborate instruwmentation, which is not warranted considering the range of property variations expecteafrom the ropes, capstans, and atmospheric effects, it is assumed that the static frictions alsoprevail in sliding and rotation.

Table 1. Calibration data of 1000-pound Baldwin tension gauge.

Gauge Tension, Gauge Tension, Gauge Tension,Reading lb Reading lb Reading lb

20 19.8 280 270.7 540 523.4

40 39.5 300 289.9 560 542.160 59.1 320 309.1 580 560.8

80 78.5 340 328.4 600 579.6100 98.0 360 347.7 620 598.4

120 117.3 380 367.2 650 626.7

140 136.6 400 386.7 700 674.1160 155.8 420 406.3 750 722.0

180 175.0 440 426.0 800 770.4200 194.1 460 445.9 850 819.5220 213.3 480 465.8 900 869.4

240 232.4 500 486.0 950 920.2

260 251.5 520 504.6 1000 972.0

Table 2. Calibration data of 4000-pound Baldwin tension gauge.

Gauge Tension, Gauge Tension, Gauge Tension,Reading lb Reading lb Reading lb

50 49.2 700 684.9 1800 1754.7

100 98.3 750 733.6 2000 1950.0

150 147.4 800 782.3 2200 2145.8

200 196.8 850 830.9 2400 2342.4

250 245.5 900 879.5 2600 2539.9

300 294.5 950 928.1 2800 2738.4

350 343.4 1000 976.7 3000 2938.2

400 392.3 1100 1073.9 3200 3139.2450 441.1 1200 1111.1 3400 3341.8500 490.0 1300 1268.2 3500 3443.7

"550 538.7 1400 1365.4 3600 3546.0600 587.5 1500 1462.6 3800 3752.0

650 636.2 1600 1559.9 4000 3960.0

S20

W-5

Errors in the computed static and sliding friction coefficients resulting from theabove assumptions are evaluated with equation 2 1. For assumed values of W = 500 pounds,Wc = 150 pounds, Fc = 8 pounds, and Ft = 15 pounds, the friction coefficient y is found

"U from the equation:

p p(110In,(W - Ft) + (Wc - Fc) (D/d)

(W - Ft) - (Wc - Fc) (D/d)"

V~ If D35 inches, IUd = 15.2 inches,

the coefficient of friction, denoted by p0, is found to be

I0O 0.52,

which is an exact value if the actual friction forces are

Fc 8 pounds,

Ft 15 pounds.

Now, if it is assumed that the actual friction force associated with sliding platform isonly 7.5 pounds, i.e., half of the statically found value, then the friction coefficient definedbyp 1 is

JAI -5 0.51.

- Therefore, yo is in error by

B 0o -# 1 0.52 -0.51 • 0.02 or 2 percent./ 1 0.51

Likewise, if the actual friction force associated with the sliding platform is

{}.= Ft 22.5 pounds,

that is, 50 percent higher than the assumed 15-pound value, then the actual friction coeffi-i cient denoted by IA2 is

2 0.53

S~21

and the error in using the friction force Ft 15 pounds, i.e., o0 = 0.51, is iAO -0A2 0.51-0.53 a5 -0.02 or -2 percent. M

In either case the error in determining the rope-capstan friction coefficient is wellwithin the limits of the 10- to 15-percent variation expected because of nonuniformities inthe rope structure, capstan surface, materials, and environment.

The effect of an error in the friction force Fc on the calculated friction coefficient IALis somewhat higher. Assuming that the actual Ft is exactly 15 pounds but the actual Fc isonly 4 pounds, the friction coefficient denoted by j3 is

"3 1 0.54

and the error in using the static friction force Fc = 8 pounds is

-0 .13 0.52 -0.540.5 - - 0.04 or -4 percent.9a3 0.54

TEST CONDITIONS IThe tests were conducted in June 1976 in a temperature range of 50 to 100 °F and a

relative humidity of 20 to 70 percent. 3For each rope-capstan combination, with the rope and capstan dry, sliding and static

measurements were alternately made three or four times for several sections of rope. The, rope and capstan were then dampened with water, and wet static and sliding tests were per-formed. Thus, the test data cover the following combinations:

conditions:

' t dry . Isliding . 1L

| static . . . . . 2 -Z

wsliding 3 TVwet. •

static . . . . . 4

In all cases, the tension weight W was maintained within 475 to 520 pounds.

Nine types of rope (table 3 and figure 11) and four capstan surfaces (table 4) weretested. ii

22

Table 3. Description of tested ropes.

Rope Material Weave Diameter, in Manufacturer

A Manila 3 strand 1.25 Columbia Rope Co.

B Dacron 3 strand 1.25 Hoover and Allison

C Polypropylene 3 strand 1.91 Danish Mfg.

D Polypropylene Plaited 1.91 Danish Mfg.

E Nylon Woven, 2/1 1.50 Samson Ag

F Dacron Braided 1.50 Samson

LG Nylon Plaitd 1.50 Columbia Rope Co.H Nacron Plaited 1.50 Columbia Rope Co.

Ij I Nylon 3 strand 1.25 Tubbs Rope Co.

Table 4. Description of capstan surfaces.

S~Surface Finish " Description '

Standard Conventional

Plasmalloy 915 (400 grit grain size emory Applied with a special plasma-jetpaper) process using a powder with the

following composition: 75%chromium carbide and 25% nickel-chrome (Cr 3C2 /NiCr)

Plasmalloy W260 (100 grit grain size emory Applied with a special arc-spraypaper) process using monel metal wire 4S

Plasmalloy W253 (150 grit grain size emory Applied with a special arc-spraypaper) process using AISI-316 stainless

steel wire

23

S- 4;

-! ý .-_ -I -4 "'1 ' -~ !!! '_;U N

00lop

IWOP 4TZ( x

F, IMjj~ AN.4 ~ -R

%LF~g Vk

7:1Part A. Danila (3 strand).

244

1 .7

44

I PPatC oyrpln 3srn)

I3

Part C. Polypropylene (plastrad).

Figure~ 11 otiud

j 2

T-~

Part. E.Nln(oe)

* AAI

-14

..............

t it

4J4

It

113

aISO

-no

Part G. Nylron (plaited).

=Figure 11. Continued.

27

T 73

tf

I¾

11

A-V

Part 1. Nylon (3 strand).

Figure 11, Continued. 4

44

AI8

TEST RESULTS

Low-Load Tests

Test results are in table 5. The last four columns provide the computed friction"coefficients for the four test conditions. The value of W in the last column is the weightacting on the tension line of the sliding platform; a reduction of 15 pounds (rolling friction)is required to obtain the actual effective tension force acting on the sliding platform.

As an example of the calculations in table 5, consider the first entry. A manila ropeand a standard capstan are used. The average of Wc for the dry static condition is

169 + 186 + 180 = 178.33 pounds.*3

The weight of the tension barrel and cable is W = 517 pounds. The dry friction coefficientfor the static condition is obtained from equation 21:

[ ttt~dr -- 1/•r In(517-15) + (178.33-8) (35/.15.2)st, dry (517-15) - (178.33-8) (35/15.2)

or

Astdry = 0.67.

High-Load Tests on Bitts

:1 For applications where the rope is subject to high tensions, the test apparatus can bemodified (figure 12) to verify the validity of the friction coefficients.

The tension forces Tl (40 to 80 pounds) and T2 (50 pounds) are applied by humanoperators. By operating the electric motor of the capstan in the clockwise direction, therope segment between the two capstans is shortened and thus stretched, which causes W toincrease. At an equilibrium point the rope around the standard capstan begins to slip, sincethe number of % rap turns around the stainless steel capstan N is greater than the number ofturns around the standard capstan n. At this point, the equilibrium equations prevail:

TI +Tý = W- Ft (23)

T*- Fc = T 1 , (24)where

= •W = tension in steel cable

Ft = friction force of sliding platform (15 pounds)

Fc = friction force of the sheave axle (8 pounds)

T', T' = tensions in rope (figure 12).

S W MThe value of 115 was disregarded because of questions concerning its validity.

29j 29

S+ ++ . ........ .. . , o. 's . . r9' A 4fl- O

Table 5. Test results.

Test Results* Coefficient of Friction

Capstan Test Run Static Sliding CIRope Surface Condition 1 2 3 4 Average** Dry Wet Dry Wet__-

Dry Static 115 169 186 180 178.33Dry Sliding 85 88 79 80 83.00 ,

Wet Static 213+ - - - 213.00 0.67 ? 0.23 0.30

Wet Sliding 98 - 98.00 W=483 W=517

Dry Static 103 120 109 115 111.75

B Standard Dry Sliding 88 93 75 75 82.75 0 .0 2 .SWet Static 98 - - - 98.00 0.

Wet Sliding 62 - - I- 62.00 W=4831 W= 517Dry Static 183+ 122 128 113 121.00

C Sa dr Dry Sliding 112 113 119 99 110.75 0.S•C Standardl 0.37 0.30 033 0.26Wet Static 103 - - - 103.00

Wet Sliding 91 - - - 91.00 W=515Dry Static 116 122 117 - 118.33

Standard Dry Sliding 107 109 107 - 107.67Wet Static 132 - - - 132.00 0.39 0AS 0.34 0.39Wet Sliding 118 - - - 118.00 ._ W=483Dry Static 89 91 92 - 90.67

Dry Sliding 67 66 67 - 66.67E Standard 0.27 0.30 0.19 0.21Wet Static 97 - - - 97.00 0

Wet Sliding 72 - - - 72.00 W=483

Dry Static 100 !01 105 83 97.25

Sd Dry Sliding 76 73 74 68 72.75 3-SF Standard 0.28 0.37 0.20• 0.24Wet Static 123 - - - 123.00

Wet Sliding 86 - - - 86.00 IW =515

Dry Static 78 85 93 - 8533 ADry Sliding 62 60 60 - 60.67

G Standard Wet Static 116 - - - 116.00 0.25 0.38 0.17 0.29

Wet Sliding 96 .- - - 96.00 W =483

Dry Static 74 82 75 - 77.00

Dry Sliding 60 60 60 - 60.00H Standard 0.22 0.27 0.17 0.16

Wet Static 89 - 89.00Wet Sliding 59 - - - 59.00 W =483

30.-

•3

Table 5. Continued.

Test Results* Coefficient of Friction

Capstan Test Run Static SlidingRope Surface Condition 1 2 3 4 Average** Dry Wet Dry Wet

-- Dry Static 213+ 187 187 - 186.75

Dry Sliding 132 138 125 - 131.70" I Standard 0.89 0.76 0.45 0.5

Wet Static 176 - - - 175.50 -

Wet Sliding 141 - - 140.50 W 480

ii Dry Static 154 165 180 181 170.00Plas- Dry Sliding 95 96 93 89 93.25malloy 0.61 ? 0.26 0.44

S! ~915 Wet Static - - - 233.00Wet Sliding 129 - - - 129.00 W=483 W;517

Dry Static 146 165 139 178 157.00Plas- Dry Sliding 109 109 106 100 106.00 '4

B malloy 0.53 0.61 0.31 0.31-• 915 Wet Static 159 - - - 159.00

- Wet Sliding 99 - - - 99.00 W=483 W=517Dry Static 169 166 169 187 172.75

C mPlaso Dry Sliding 119 125 133 126 125.75•.C ,Malloy We ttc 19 - - -0.63 0.87 0.39 0.40

915 Wet Static 199 - - - 199.00Wet Sliding 128 - - - 128.00 W=515Dry Static 198 174 179 180 182.75

Plas- Dry Sliding 129 137 137 130 133.25D malloy 0.75 0.67 0.44 0.27D 915 Wet Static 173 - - - 173.00

Wet Sliding 93 - - - 93.00 W = 500

Dry Static 157 166 175 164 165.50Plas" Dry Sliding 98 92 91 94 93.75

E malloy 0.61 0.67 0.27 0.28915 Wet Static 173 - - 173.00

Wet Sliding 94 - - - 94.00 W =500t Dry Static 154 161 148 157 155.00 PPlas- Dry Sliding 124 111 99 103 109.25

F malloy 0.52 0.63 0.32 0.35915 Wet Static 172 - - - 172.00

Wet Sliding 118 - - - 118.00 W=515

Dry Static 153 160 161 158 158.00G Plas- Dry Sliding 100 104 99 104 101.75

o"G malloy 0.54 050.9 0.31915 Wet Static 166 - - - 166.00

___ Wet Sliding 105 - - - 105.00 W 515

31x

1t

7.2

Table 5. Continued.

Test Results* Coefficient of Friction

CRun Static Sliding "SCapstan TestRope Surface Condition 1 2 3 4 Average** Dry Wet Dry Wet

Dry Static 162 151 173 181 166.75Plas- Dry Sliding 94 87 93 92 91.50

H malloy 0.62 0.94 0.27 0.29915 Wet Static 198 - - - 198.00

Wet Sliding 98 - - - 98.00 W= 500

Dry Static 156 150 148 - 151.00Plas- Dry Sliding 117 111 114 - 113.80 0

I malloy 0.56 0.46 0.37 0.62

915 Wet Static 133 - - - 133.00

Wet Sliding 160 - - - 160.00 W= W480

Dry Static 162 184 136 123 151.25Plas- Dry Sliding 103 106 93 88 97.50

A malloy + 0.53 ? 0.29 0.25W253 Wet Static 207 - - -

Wet Sliding 88 - - - 88.00 W = 500

Dry Static 218 182 179 164 185.75Plas- Dry Sliding 120 113 111 112 114.00 ' 0. 03B malloy WtSai 20+ -0.78 0.35 0.35

W253 Wet Static 202 - - -

Wet Sliding 113 - - - 113.00 W=500

Dry Static 177 167 167 176 171.75Plas- Dry Sliding 121 130 120 121 123.00

C malloy 0.67 0.71 0.39 0.41W253 Wet Static 176 - - - 176.00

Wet Sliding 126 - - - 126.00 W=4960uDry Static 178 173 155 146 163.00

D P'al. Dry Sliding 120 123 119 125 121.75 03 .D malloy 0.61 0.93 0.39 0.38 •W253 Wet Static 196 - - - 196.00

Wet Sliding 121 - - - 121.00 W=496

Dry Static 219 190 160 172 174.00Plas- Dry Sliding 122 117 119 112 117.50 0 . . .

E malloy 0.70 0.77 0.37 0.39 ••_

W253 Wet Static 181 - - - 181.00

Wet Sliding 121 - - - 121.00 W=492

Dry Static 163 164 163 158 162.00Plas- Dry Sliding 101 130 125 136 123.00

F malloy 0.64 0.55 0.41 0.45W253 Wet Static 149 - - - 149.00

Wet Sliding 131 - - - 131.00 W=480

32

Table 5. Continued.

Test Results* Coefficient of Friction

• • Capstan Test _Run Static Sliding

Rope Surface Condition 1 2 3 4 Average** Dry Wet Dry Wet

Dry Static 206+ 209k+ 220+ - 211.67+

Plas- Dry Sliding 149 163 156 - 156.00G inalloy ? ? 0.59 0.59

W253 Wet Static 222"r - - - ?W5 Wet Sliding 156 - - - 156.00 W=483

Dry Static 173 196+ 201+ - 173.00.las- Dry Sliding 116 122 117 - 11833H• malloy +0.72 ?• 039 0.44

Wet Static 209 - - -

Wet Sliding 130 -. . W =483Dry Static 169 182 176 - 175.70

Plas- Dry Sliding 132 124 130 - 128.70I malloy 0.78 0.70 0.44 0.60

W253 Wet Static 170 .- 170.00

Wet Sliding 156 - - - 156.00 W=480

Dry Static 162 15' 97+ - 156.50P"as- Dry Sliding 112 87 89 - 96.00

A malloy + 0.60 ? 0.30 0.24- W260 Wet Static 232 - - -

Wet Sliding 82 - - 82.30 W=480Dry Static 189 .62 14 - 165.00

Plas- Dry Sliding 126 113 111 116.67B malloy 0.66 0.53 038 0.32

? B ma26o Wet Static 145 - - 145.00W260 WeI

Wet Sliding 103 -.. . 103.00 W=480

P Dry Static 160 130 138 - 142.67Plas- Dry gliding 122 125 132 - 12633

C malloy 0,5 1 0.70 0.43 0.40W260 Wet Static 170 - - - 170.00

Wet Sliding 120 - - - 120.00 W=480

Dry Static 172 148 152 - 15733

Pls Dry Sliding 159 144 150 - 151.00 5D malloy 0.60 0.56 0.56 0.40

Wet Static 151 - - - 151.00SW260Wet Sliding 121 - - - 121.00 W 480

Dry Static 181 167 127 - 158.33maloy Dry Sliding 116 117 105 - 112.67

W260 Wet Static 178 - - - 178.00Wet Sliding 109 - - - 109.00 W 480

33

NA,-1

Table 5. Continued. I-Test Results* Coefficient of Friction

Capstan Test Run Static Sliding

Rope Surface Condition 1 2 3 4 Average** Dry Wet Dry Wet

Dry Static 153 146 160 - 153.00 IPlas- Dry Sliding 115 120 111 - 11533

F malloy 0.57 0.50 0.38 0.37W260 Wet Static 141 - - - 141.00-,

Wet Sliding 113 - - - 113.00 W=480

Dry Static 162 172 162 - 165.33Plas- Dry Sliding 111 117 121 - 116.33

G malloy 0.66 0.64 0.38 0.38W260 Wet Static 162 - - - 162.00 I

Wet Sliding 116 - - - 116.00 W=480

Dry Static 182 194 172 - 182.67Plas- Dry Sliding 132 112 110 - 118.00

H malloy 0.83 0.62 0.39 0.38W260 Wet Static 160 - - - 160.00 _ _

Wet Sliding 116 - - - 116.00 W=480

Dry Static 151 132 161 - 148.00Plas- Dry Sliding 108 114 103 - 108.30

malloy + 0.54 0.35 0.35 J,W260 Wet Static 193 - - - 193+

Wet Sliding 110 - - 110.00 W=480

*Values for We, Le., the weight of the barrel, its contained water, and the vertical segments of the steel cable.

"**Arithmetic averagL of test data.

+Data disregarded in averaging process because of concern for validity.

From the equilibrium equations, the tension T' can be found:

Tl = (1/2) (W- Ft - Fc) = (W - 23)/2. (25)

The tensions T1 and T1 are related by equation 11:

T• = T1 exp (par), (26)

where

oz = n X 27r radians

IA = sliding friction coefficient between rope and standard capstan. fl

34

Sbitt (fixed axis, friction test surface) stainless steel ca Dstan

jrope TK T2"• • • N turns

1 in~winch

tsh sliding platform betwee n turnsstandard

a s s e ecapstan tension- gauge W

•' T1

"• ~steel

Ui=U

floor

•Figure 12. Modified apparatus for high-tension rope tests (or bitts).

! { From equations 25 and 26, the friction coefficient y is found:

1 In (W- 23)/2 (27)i• P -2 n r " Ti ..

where T I is within the range of 40 to 80 pounds.,!!, The ropes A, B, E, F, G, and H were tested with two pairs of values for n and N. In

each test the tension W, which causes the sliding condition between the rope and the stand--,: •ard capstan, was recorded; table 6 shows the results.

Sii 35-

Table 6. Results of high-load tests and related calculations.

Wrap Turns Tension Friction Coefficients

Rope n N (W), lb High Load* Low Load**

A 2.25 3 5500 0.30 to 0.25 0.23

B 2.25 3 1400 0.20Oto0. 15 0.23B 3.25 4 5500 0.21 to 0.17 0.23-45

E 2.25 3 600 0.14 to 0.09 0.19

E 3.25 4 4000 0.19 to 0.16 0.19

F 2.25 3 1500 0.21 to 0.16 0.2C

F 3.25 4 3000 0.18 to 0.14 0.20

G 2.25 3 1500 0.21 to 0.16 0.17G 3.25 4 3400 0.18 to 0.15 0.17o•

H 2.25 3 1500 0.21 to 0. 16 0.17P

H 3.25 4 4000 0. 19 to 0. 16 0.17

*High4load friction coefficients are computed by using equaton 27 with T, first replaced by 40 and then by 80 pounds.

"*Lowload friction coefficients were transfenedfrom table 5, drysliding column, for tire ropes A, B, E, F, G, and H onstandard capstan. i

APPARATUS AND PROCEDURES

Bitts or bollards are short pieces of heavy gauge, large diameter pipes installed on thedeck with safety plates welded to their upper ends to prevent the rope from slipping off.They are used extensively in towing and mooring, and the friction coefficient of ropes onbitts plays an important role in the safety and smoothness of these operations. Realisticdata on rope-bitt friction coefficients are also needed to develop effective standard proce-dures for the use of synthetic fiber ropes. T!j

The friction coefficients obtained from capstan tests apply to some extent to bittsor bollards. However, a new set of tests directed towards determining the effect, if any, ofbitt diameter was necessary. The apparatus used was the basic test setup (figure 6) with thefollowing modification-:

The test capstan was removed, and pipe segments of desired diameters (simulatingthe bitts) were welded to the frame (figure 13).

The sliding platform was anchored across a tension gauge G to the frame.

36

11

brrel

theoright edo

-The test rope was wrapped several turns around the capstan, and an operator held

SThe left end of the test rope was wrapped 1.25 (or 2.25) turns around the bitt, and-j thnattached to a barrel containing water (100 to 500 pounds). Thus, a constant ,rT 2 equal to the total weight of the barrel was applied to the left-end of the

By using this apparatus, a clockwise rotation of the capstan by its electric motor hlj ;~~~~, ~the left; a nd cuethtnsonfi the aoe n chor ln idctdbh tension.gue )t

shortens the line segment between the capstan and the bitt; moves the sliding platform to;. j the left; and causes the tension T in the anchor line (indicated by the tension gauge G) to "

increase. As the tension T increases, the tension T 1 transmitted to the line leading to thebitt, which is given by the equation

T = T + (friction force of piatform roller3),

is increased. Thus, as the drive motor continues to rotate clockwise, eventually the tension AT, overcomes the rope-bitt friction force and causes the rope to slide around the bitt.

*k[3 While the rope is sliding, the tensions T, T 1, and T2 remain constant, and the follow-ing equations are valid:

T1 T+Ft (28)

37

I

and

T = T, -I.p (as, a), (29)

whereIi Ft = friction force of platform rollers

p 1s = sliding friction coefficient of rope on bitt surface

4 ao = wrap angle of rope around bitt in radians.

In the above equations, T is known from the tension gauge reading; Ft 15 pounds(determined experimentally); ae is known from the rope's wrap angle (1.25 or 2.25 turns);and T2 is the total weight of the barrel. Therefore, the only unknown is the sliding frictioncoefficient psl which is computed from the following equation:

I T+Ft

ps -n Tn (30) 1! /as = ot T2

Immediately before the rope starts to slide around the bitt, the tension T, indicatedby the gauge G, reaches its maximum value Tmax. The static friction coefficient plst can al- -*so be computed from equation 30 by replacing T with Tmax, i.e.,

1 Ini!2 Tmax + Ft (31)Pst a T2 (31)

To obtain accurate values for Tmax, lowever, the tension T must be increased very slowly,something that cannot be done with the present motor-driven capstan. {TEST DATA AND COMPUTED FRICTION COEFFICIENTS

Two bitt diameters were tested: 8-5/8 and 16 inches. With each diameter severaltypes of ropes were used, and with each rope-bitt combination two wrap !urns were used:ee1 = 1.25 turns = 1.25 X 2 ir = 7.85 radians and a2 = 2.25 turns = 2.25 X ir = 14.14 radi-ans. The test data are shown in table 7. Since the surface roughness of an actual bitt barrel

___ Table 7. Bitt test data and computed friction coefficients.

Rope Pipe Wrap-Tl ,

Rope Diameter, in Turns T2, T lb T, lb** 72"

200 700 0.16Manila, 1.25 400 1480 0.173-strand, 8.625 0.15 0.161.25 in 2.25 200 1530 0.14

38

m3

Table 7. Continued.

Pik.e WrapSRope Diameter, in Turns T2, lb T, lb JA* l * 2t

ISO 700 0.19Manila, 1.25

300 1220 0.183-strand, 16.0 - 0.17 0.16

150 1530 0.161.25 in. 2.25300 3300 0.17

200 910 0.191.25400 1550 0.17

8.625 0.16Dacron, 2.5 200 1770 0.152-in- 1 400 2940 0.14

0.19woven, 150 1000 0.241Tb7.Ctne1.25

15in300 1720 0.2216.0 - 0.21150 620 0.20

2.25 300 4000 0.18

200 820 0.181.25 400 1270 0.15

8.625 0.152.5 200 1440 0.14Dacron, 2.25 200 1770 0.15

plaited, 400 2940 0.140.16150 640 0.181.50 in 1.25

, 300 1270 0.1816.0 0.17150 1810 0.18

2.25 300 2625 0.15

200 840 0.181.25 1 400 1450 0.16

8.625 0.15

8.lon 200 1770 0.15

Nylon, .252-in-,1 2.400 2920 0.14 0.16

woven, 150 1060 0.251.50in 1.25 300 1400 0.19

16.0 0.21150 2840 0.21

2.25 30 265 01

300 4030 0.18

200 1060 0.21o ,, '1.25 1.25 400 1420 0.16

8.625 0.16S200 1480 0.14- Nylon, 2.25

2-ni400 2860 0.14| _.__plaited, 0.161.50 in 1.25 150 730 0.201300 1000 0.15

16.0 0.216

--- 2.25 150 1330 0.152.25 200 1060 0.1

3 00 1480 0.14

Nylon ~ 40 286 0.1 0.1

Table 7. Continued.

ST Pipe WrapRope Diameter, in Turns T2 , lb T, lb pU 1* 2t

1.25 200 860 0.181 400 1350 0.158.625 0.15

Nylon 2.25 200 1530 0.14

3-strand, 400 2630 0.13 0.171.25 in 1.25 150 830 0.22 I300 1280 0.1816.0 10..18 0.12.25 150 1770 0.17

300 2950 0.16

"*Friction coefficient for each test.

**A verage friction coefficient for each pipe diameter.

tA verage friction coefficient for each Pope assuming no diameter.

is similar to that of a pipe segment (rather than a capstan), the values of the friction coeffi-cients in table 7 are more realistic for bitt barrel applications than those in table 5.

The average friction coefficients for larger pipes are generally higher than those forsmaller pipes, since the tests with larger pipes were made after a two-day storm had com-pletely soaked the lines, a factor which resulted in higher frictions (see table 5 for similareffect in dry versus wet coefficients for capstans).

APPLICATIONS

INTRODUCTION

This section discusses how data developed in the preceding sections can be used formarine applications of synthetic ropes. Such applications include the following:

. Line loading effect on bitt barrels and resulting stresses

. Line stretching effect• Wrap turns required in using capstans

• Effectiveness of vwious line wrapping configurations in mooring and towing

-40

-i

N1

;, •ISTRESSES IN BITTS AND BOLLARDS

The most commonly used, welded type bitts are assembled according to the Bureau

of Ships' drawings 805-921985 and 805-1361959 (figuie 14).* Nominal values for somedimensions are in table 8.

' In mooring and towing operations, the bitt is subjected to the tension force of the rope- which is wrapped around it. Considering a single turn of the rope, the tension forces are in a

plane nearly perpendicular to the axis of the bitt. These forces can be resolved into a forceF = T1 - T2 passing through the axis and a twisting moment Mt = (TI -T 2 ) (Do/2) passingaround the axis of the bitt (figure 15).

When the rope is wrapped several turns around the bitt, this method of resolutioncan be separately applied to each turn. However, since the resultant forces generally inter-

sect the axis at different angles and are located in different planes passing through the axisof the barrel, the method is only a time-saving approximate technique.

If the single turn of the rope around the bitt, considered for this discussion, is at aheight h above the base of the bitt, then the force F and the twisting moment Mt are in theplane containing the turn (figure 16). The force F results in (1) bending of the barrel in theplane of its axis, causing longitudinal fiber stresses (tensile on one side and compressive onthe opposite) and (2) transverse shear stresses parallel to F. The twisting moment Mt causestorsional rotation of the barrel and torsional shear stresses within the barrel. The higheststresses are developed at the base of the barrel where deformations are restrained by thewelded plates.

*805.921985 is used for two-barrel bitts (reference 2) and 805-1361959 for three-barrel bitts (reference 3).

Table 8. Nominal dimensions for two-barrel bitts (reference 2).

Top-PlateDianr.-ter Thickness Barrel Spacing Height Diameter

(D), in (T), in (B), in (C), in (d), in

4.500 0.337 9.0 12.0 6.000

6.625 0.432 13.5 13.5 8.625

8.625 0.500 18.0 15.0 11.125

10.750 0.625 26.0 16.5 13.750

12.750 0.625 30.0 18.0 15.75014.000 0.750 36.0 19.5 17.000

S16.000 0.750 42.0 21.0 19.000

18.000 0.875 46.0 22.5 21.00020.000 1.000 54.0 24.0 23.000

41

B

D- t I C

front elevation and section

1 K]

bottom view |

•','""°'••"" •,,':Figure 14. Mooring bitts, two-barrel welded type. .

442

T2€ T I1

Figure 15. Resolution of rope tension forces acting on bitt drum.

AA I F

tM

SFigure 16. Forces acting on a bitt drum by a single turn nf ropeat height h above drum base.

"43

M-

The maximum fiber stress at the base of the bitt caused by the bending moment

M = Fh (32)

is given by the following equation (reference 4):M (33

°max 2 I/Do'

where I denotes the moment of inertia of the barrel's cross-sectional area with respect to theplane of neutral axis and DO is the outside diameter of the barrel. Since

0V

S-4), (34)

equation 33 can be written as -J

32DoM DUmax (D D4) (35)

The maximum transverse shear stress at the base of the bitt (reference 5) is

Tvmax = 2V, (36)

where V=F is the shear force at the base and I4 A (' o il

is the cross-sectional area of the barrel. Therefore, equation 36 can be written as

8= (22 ) F. (37) 7S7r (D -D .

The maximum torsional shear stress at the base of the barrel is given by the follow-

ing expression (reference 4): {16 Mt Do (38)

Tt,max= (4_D4)

When the line is wrapped several turns around the bitt, it is necessary to plot dia-grams for the transverse shear stress and bending and twisting moments along the height of 4S~tie barrel to determine Fmax (the maximum shear force), Mmax (the maximum bending

moment), and Mt,max (the maximum twisting moment) for use in equations 35, 37, and 38.

In present applications, low alloy, high-strength stainless steel, e.g., A242 - 63T, isnormally used for construction of bitts. The material suggested in reference 2, which isoften used for bitt barrel construction, is commercial steel (ASTM A7-61 T) for which therecommended allowable working limits are

44 .~. L

S°max - 20,000 pounds per square inch

Tv, max = Tt,max =15,000 pounds per square inch.

Using these values in equations 35, 37, and 38, the allowable limits of F, M, and Mt (denot-ed by Fag Ma, and Mta, respectively) can be found:

Fa =ir (D -D') 5(8 - Tv,max o5890 (Do2-D2) (39)

4 44Ma = 032Do max =1963 (D4-D4Do (40)

0] Mta lD t,max = 2945 (D4 - D 4) /Do (41)ta 16 Do 0ta oiI

.1 • Equations 39, 40, and 41 provide the upper limits for structurally allowed loads for a bittbarrel of given Di and Do.

2i mateIn table 9, the values for Fa' Ma, and Mt,a are listed. These limits are only approxi-mate, since the combination effects of various stresses have been disregarded.

Table 9. Allowable limits of shear force, bending moment, and twisting moment for bitt"j -barrels made of commercial steel (ASTM A7-6 IT), Omax 20,000 psi,

Tt,max Tv,max = 15,000 psi.

Do Thickness Fa, Ma, Mt,a,in* ((Do-Di)f2), in* lb X 103 in-lb X 103 iaz-lb X 103

4.500 0.337 33 85 128

6.625 0.432 63 244 367

8.625 0.500 96 490 73510.750 0.625 149 951 1427

12.750 0.625 179 1376 2064

14.0'00 0.750 234 1963 2945

takn 0.750 269 2617 3926i.(i;0.875 353 3844 5767

20.000 1.000 448 5400 8102

"*Nominal values taken from table 8.

45 '

Example 4

In an application, the bitt barrel is subject to tensions T1 = 40,000 pounds andT2= 1000 pounds of a line which is wrapped one turn around the barrel (figures 15 and16). The problem is to determine if the structurally allowed limits of loading are exceeded. _

The dimensions are Do = 10-3/4 inches, thickness = 5/8 inch, and h = 10 inches. Data intable 9 provide the following values for the allowable shear and moment limits:

Fa= 149,000 pounds,

Ma= 951,000 inch-poundsM a

Mta = 1,427,000 inch-pounds. -"

The maximum shear, bending, and twisting moments are

Fmax 39,000 (<149,000) pounds,

Mmax = Fh = 390,000 (<951,000) inch-pounds,

Mtmax F Do/2 = 209,625 (< 1,427,000) inch-pounds, -

Therefore, the allowable loading limits of the bitt have not been exceeded: 4.

Fmax 26 percent of Fa ti

Mmax 5 50 percent of Mai Mt~max 5 15 percent ofMta

Example

In an application, the bitt barrel is subject to tensions from two turns of a linewrapped around the barrel as shown in figure 17A. The bitt barrel's outside diameter is10-3/4 inches and its thickness is 5/8 inch; the magnitude and direction of the forces andtheir locations are as given in figure 17B. The problem is to determine if any of the allowa-ble loading limits has been exceeded.

Based on the diagrams for the transverse shear force, bending moment, and twisting

moment (figure 18), it can be concluded that

Fmax = 39,000 pounds, IMmax = 39,000 X 14-950 X 8 = 538,400 inch-pounds,

Mt,max 39,000 X 1 + 950 X 1075 214,731.25 inch-pounds

46

T3.

bitt

free to rotate T1 i0b

T3 = B~

Part A. Arranlgement of rope 'urnls

F2 hiI l I i

3900. 1 14i

S+free to ro0t -i" 00Ib !

Part B. For cesandm n rp twist n s. M

a c in o bi t b arr- --- +

aFtiguo it re 1. Bi tb -Te u j c " e so

:+.+-++'. i :47

CR,

14

I -l

2x10 5 2x10 4 shaI, .\ j shear.

twisting moment bending moment

12 ii

10

F2 950

8 Mt2 5106.25,\-1

Mt = 214,731.25 M = 234 000 'A AS

F = 38,050

6 \

I

2

I \I-

M 538,400Mmax,

0 LFreaction bitbase 20105x10

Mt, reaction

Figure 18. Shear, bending, and twisting moments for bitt barrel subject to tensions ftrom two turns of line. *

48

!E

From table 9, the allowable limits are obtained:

Fa = 149,000 pounds,

j Ma = 951,000 inch-pounds,

Mta = 1,427,000 inch-pounds.

Comparison of maximum leads with the allowable limits indicates that neither limit hasbeen exceeded.

ROPE ELONGATION EFFECT

eralyUnder tension, synthetic lines in general and nylon lines in particular stretch consid-Serably more than the natural fiber lines. A stretch of 33 percent of the rope length under a"safe working load is quite natural in a nylon line.

In marine applications, particularly during high waves, tension varies periodically.As it increases, elastic stretching of the line increases, and as it lessens, the line returns to itsrelaxed length. This phenomenon can cause excessive wear in the sections of the line which,in the course of line expansion and retraction, rub against the bitt surface. In the case of"chromium carbide plasmalloy" or "stainless steel arc-spray" bitt surfaces, this wear cancause excessive localized weakness and eventually premature failure.

The following example is presented to provide some numerical values on the extentof rope travel over the bitt because of elongation. The effect on the rope's life cannot bequantitatively estimated at this time because of lack of experimental data.

Example

A 12-inch-perimeter nylon line is wrapped several turns around a bitt barrel (figure19). The high tension acting on the line varies periodically from 10 pounds (relaxed condi-tion) to 40,000 pounds (tha safe working load). The problem is to determine the maximumtravel amplitude resulting orom stretching the line segment that is wrapped around the bar-rel. The outside diameter of the barrel Do is 10-3/4 inches the tension at the relaxed end ofthe line is very small, and the rope-bitt friction coefficient J is 0.4.

The equation for the tension along the line segment which is wrapped around the

bitt is

T T1 exr, -u-c•),

where cc is the wrap angle measured from point of contact A. With T1 = 40,000 pounds andA = 0.4, a tension of 10 pounds prevails at

1 -1 n 40 '00 0--] In(T I/T) 0 .4 Into10

20.735 radians or 3.3 turns.

J

49

L , a' 51

bitt

A iT

'fr-

Figure 19. Description of conditions and symbols; wrap angle A,B is 3.3 turns;TI varies periodically between 10 and 40,000 pounds.

Using the above wrap angle, point B (at which the line tension is 10 pounds) is located.Since the tension in segment B-C is always less than 10 pounds, the resulting stretch is negli-gible and it is possible to concentrate on segment A-B. To simplify the equations, it is as-sumed that the line tension at B is constantly maintained at 10 pounds. The expression forline tension in segment A-B is

T T2 exp (0.40) = 10 exp (0.40) (42)

where 0 is the variable wrap angle measured with respect to B. At B, where 0 0, jT = 10 exp (0.4 X 0) = 10 pounds (as assumed),

and at A, where 0 =20.735 radians,

T = 10 exp (0.4 X 20.735) -: 39,998 (• 40,000) pounds

which is the maximum external tension applied at A.

To express the line elongation under tension, it is assumed that the stretching iselastic and linear with respect to tension in the range of 10 to 40,000 pounds, i.e.,

:.1 ---•-AL (43)

a 40,00 (4350

where AL is the elongation of a section with the relaxed initial length L under applied ten-sion T and a is the coefficient of linearity.

If we assume that a tension of 40,000 pounds results in a fractional elongation of"0.33, i.e., AL = 0.33 L,

T! 040,000i! ~0.33 =a X 4,040,000

or

a 0.33

(from equation 43). Thus, the strevh-tension expression becomes

AL = LT. (44)40,000

Equation 44 is now applied to an infinitesimal section of the wrapped rope. In this

case, the infinitesimal length L can be expressed in terms of an infinitesimal wrap angle dOby the expression:

:*L =(Do/2) d0 (45)

Substitution of equations 42 and 45 into equation 44 results in the expression:

0.33 Do: , AL = 4 10 exp(0.40) dO, (46)

which defines the 2longation of an infinitesimal section at location 0.

Integration of equation 46 with respect to 0 from 0 = 0 (point B) to 0 = x (a movingpoint M within segment B-A with wrap angle x) provides the displacement of M as a result of

T1 ••stretching of length B-M of the rope. Thus 3

M 0.33 10.75 [e_0"4]- Y--1~Y [L] =4 0 0 10

B 40,00 20.4

or

"Y 1.109 X 10- 3 (eO.4x-1), (47)

It where Y denotes the stretching of the B-M segment and thertiore the displacement of pointM over the bitt surface under tension T1 =40,000 pounds.

-~ 51

$

Numerical values of Y for values of x ranging from 0 to 20.735 are in table 10, anda plot of Y as a function of x is in figure 20.

As shown in figure 20, the displacement of the rope over the bitt at point A (figure19) under the elongation effect of tension T1 -= 40,000 pounds is almost 4-1/2 inches. Fur-thermore, figure 20 shows that from x = 4.27r to x = 20.735 the rope's displacement over thebitt's surface is greater than 0.2 inch. Therefore, if it is assumed that 0.2 inch of displace-ment is large enough to cause a chafing effect on the rope by rubbing it against the bitt sur-face, the line segment wrapped around the barrel from x = 4.27r to x = 20.735, i.e., a lengthof

(Do/2) (20.735 - 4.27r) = 10.75 7.54 40.5 inches,

will be affected. Consequently, under repeated tension and relaxation, a weak section 40.5inches long will develop in the line. This is an area that warrants experimental investigationfor better prediction of localized wear, and thereby development of safe, standard, operatingprocedures.

Table 10. Rope displacement over bitt barrel because of stretchingas a function of angular location around bitt barrel.

x, rad Y, in

1. 0(point B)

0.57r 0.0010

1.07r 0.0028

1.5•r 0.0062

2.-0r 0.0126

2.5ir 0.0245

3.Oir 0.0470

3.57r 0.0890

4.07r 0.1678

4.5ir 0.3156

5.Oir 0.5925

5.51r 1.1116

6.Oir 2.0847

6.57r 3.9087

20.735 4.4330(point A)

I~I 52•::5:: 5

I

5[4ii,

z

U-.

I LU

to a

-J

0 2

z

J -j• 0

W M

0.

11. ______ _.

2 3 4 5 6 7

LOCATION ANGLE (x), rad

Figure 20. Displacement over barrel because of line stretching as a function of location angle (equation 47).

"1.• •II53

FIR,

HOISTING, MOORING, AND TOWING

In this section four examples are presented: one on the use of capstans and gypsiesfor hoisting and three on the use of two-barrel bitts for mooring and towing.

Example

A motor-driven capstan and a 12-inch-perimeter, dacron, three-strand line are used tozj apply a safe working load of 40,000 pounds. The problem is to determine the minimumwrap angle (turns around the capstan) required for various capstan surface finishes. Assumethat the low-tension end of the line is subject to a 50-pound pull (figure 21).

The equation relating T1 , T2 , p, and a is

T1 = T2 exp(p ox) (T 1 > T).

Knowing T1 , T2 , and p, the above equation provides the wrap angle, i.e.,

,T,

I-n- (48)T,),

;~ T2:50 lb

T1 40,000 ib

motor-drivencapstan

Figure 21. Use of capstan in hoisting.

54 A

'44

K- or the number of wrap turns

n -- In (T/T2) (49)

From test data on the static friction coefficient of 1.25-inch-diameter, three-strand,dacron rope, on various capstan surface finishes, the values in table 11 are extracted.,

Table 11. Values for rope B.

"Standard Stainless Plasmalloy MonelCapstan Surface Surface Steel 915 Metal

,i (dry, static) 0.33 0.78 0.53 0.66

Using these values in equation 49, with T1 40,000 pounds and T 2 50 pounds, the valuesS I in table 12 are computed.

Table 12. Wrap turns required for pulling 40,000-pound load with a three-strand,12-inch-perimeter, dacron line under dry condition with various capstans.

Standard Stainless Plasmalloy MonelCapstan Surface Surface Steel 915 Metal

Minimum wrap, 3.22-4 1.36-,2 2 1.6 =, 2n (turns)

Safe wrap, n 7 3 4 4(safety factorof 2)

4', ,

Example

In a lead-to-tow application of a two-barrel bitt, "figure-eight" wrapping configura-

- tion (figure 22), the high-tension force is 40,000 pounds. The problem is to determine theI tensions along the wrapped line. The line is a 12-inch-perimeter, three-strand, dacron line

and the bitt barrel has a standard surface finish. The geometrical dimensions and distancesare shown in figure 22.

From table 5, the coefficient of friction for a dacron line (assumed independent ofLi )line diameter) against a standard capstan surface (now applied to the bitt barrel) is

A= 0.33.

barrel 2 barrel 1J

~T7

183 inn

¶1 3

6-3/-3/in\

Fiur 2.Fgue~ihtlnecnfguaio rondto-are itt intwig2

Figuwrap anFglresegtln ofgrto aroundth two-barrels for coseutv sgetsoftewrappe

lini: starting tfrom the high-tension end, are, respectively,

(ty1 -t- 27r). (a, + 27r+ a,), (a, + 21r+ 2a)) ,

(aI + 27r+ 3 2 ), (a I + 27r+ 4a,))

(a, + 27r+ 4a2 +c 1l + 27r) ,

4 ~where '

a11 sin- 1(6.375/15) ý_-25 degrees 0.43896 radian,

o 2̀ =2a, + 180 degrees 230 degrees 4.0 radians.

Eli56

[1 '

Using the tension equation,

T = T1 exp(-ga) ,

where T, = 40,000 pounds, tensions at various locations along the wrapped line segments

are computed. Thus,

T = T 1 exp[-p/(aI + 27r)] - 4352 pounds

T3 = T1 exp[-p(ot1 + 2r + ca2)] = 1162 pounds,

T4 = T1 exp[-p(aI + 27r + 2a 2 )] a5 31C pounds,

T5 = T1 exp[-jiax 1 + 27r + 3o12 )] ! 83 pounds,

T6 = T1 exp[-ju(at1 + 27r + 40'2)] a 22 pounds,

"T7 = T1 exp[-M(a 1 + 27r + 4a 2 + a1 + 27r)]1- 2.4 pounds.

(refer to figure 22).

To determine if the maximum loading limits have been exceeded, barrel I is con-sidered. The locations of resulting forces and twisting moments are assumed to be as infigure 23A. The transverse shear stress, bending moment, and twisting moment are infigure 23B.

From figure 23B, the following maximum values for shear force F, bending momentM, and twisting moment Mt are obtained:

Fmax = 34,569 pounds,

max =101,095 inch-pounds,

f {Mt,max = 234,135 inch-pounds.

Structurally allowed limits for a bitt barrel of 12-3/4 inch outside diameter are

Fa = 101,000 pounds,

' I Ma = 1,135,000 inch-pounds,

Mt,a = 1,169,000 inch-pounds.

(from table 9).

57

barrel 1

F 3 =310- 83 227 b - Mt 1= 1447 in-lb3/

F 2 =1162-310=8521b Mt, 2=5432in'lb

Mt 3i 227,256 in-lb

5in F= 40,000 -4352 35,648 lbS~3in

7777, // ,' ,17/ /

Part A. Forces and bending moments.

h, in

7 Mt =1447 F =-227

I;Mt =6879 F=-1079 r M=454shear

twisting moment,,,,

Mtax = 234,135

Mmax= 101,0 9 5

S2 g = m 34,569

Part B. Shear and moment stress values.

Figure 23. Forces and moments for figure-eight line configuration around two-barrel bitt in towing.

58

~ £1

Since Fmax < Fa, Mmax < Ma, and Mt,max < Mt,a, it can be concluded that nostructural limit has been exceeded.

As discussed earlier, this method of developing diagrams for shear stresses andmoments necessitates the assumption of coplanarity for all forces F1 , F2 , ... , which maynot be true. Therefore, the method can be considered as a time-saving approximate tech-nique which in most cases provides an upper limit for the shear and moment stresses.

Example J

"Wrapping the line around the bitt barrels follows the configuration shown in figure41 ;24. All dimensions, rope properties, and applied high tension remain as in the previous

example. The problem is to determine the tensions along the wrapped line.

The rope angles around the two barrels for consecutive segments of the line separat-ing tensions T1 and T2 , T2 and T3 , and T3 and T4 are, respectively, 31r, 37r, and 47r. Usingthe equation:

T = T1 exp(-po),

barrel 2 barrel 1

3T3

__ -- _,- _'

_ --,-T2

• I

IJ I

__ --

11I I

T3 T1'

T4hiT

Figure 24. Nonreversing-wrap line configuration around two-barrel bitt in towing.

59

whereT = 40,000 pounds and j= 0.33, the tensions along the wrapped line can be Icomputed:

T2) = 40,000 exp -0.33(37r) --- 1784 pounds, VT3= 40,000 exp[-0.33(3ir + 31r)] a5 80 pounds,

T 40,000 exp[-0.33(37r + 37r + 47)] - 1.25 pounds.

The forces and twisting moments are assumed to be at the locations shown in figure25A; the resulting shear, bending moment, and twisting moments are in figure 25B. The

maximum values for the transverse shear and moment stresses are ootained from figure 25B:

Fmax = 38,295 pounds,

Mmax 153,417 inch-pounds, .

Mt,max 244,131 inch-pounds.

"Comparison of these values with the allowed loading limits for this size bitt barrel (table 9),i.e.,

Fa 101,000 pounds,

Ma =1,135,000 inch-pounds,

Mt,a= 1,169,000 inch-pounds,

indicates that no structural limit has been exceeded. However, the magnitude of loads inthis configuration is generally higher than those in the figure-eight configuration in the pre-vious example.

Example

Tne wrapping of the line around the bitt barrels for a lead-to-tow application is asshown in figure 26. The problem is to determine the tensions along the wrapped line. Prop-erties, dimensions, and pull forces are the same as in previous examples.

Based on data in figure 26, the wrap angles around the two barrels for consecutive* segments of the line separating tensions T I and T2, T2 and T3 , T3 and T4 , and T4 and T5

are (37r + t), (31r + 2otl), (3wr + 2ci), and 27r, where

I= sin-l(6.375/15) - 25 degrees a 0.439 radian.

60S.. .. . . .

barrel 2

2 e80 -1=79,b

I Mt, 2= 504

7 in

F 40,000

- = 38,216 9b

44in

M j t n eMt, R

Part A. Force and twisting moments.

h, in

Mt =54 F =79

/,she,,Mt =244,131 iF =38,295 '

Jtwisting moment-"

.,- jbendingI moment----v-%-•

SJ • Mmax= 153,417JtMrma= 244,131••

i: Fmax =38,295•

i'll Part B. Shear and moment stress values.

Figure 25. Forces and moments for nonreversing-wrap line configuration around two-barrel bitt in towing.

61

3~1t t

barrel 2 barrel 1

I-I I

•iT4

_T2 " T1

TT

Figure 26.. Nonfigure-eight reversing-wrap line configuration around two-barrel bitt in towing.

By using the tension distribution equation]

T =T 1 exp(-pa)

the tensions along the wrapped line can be computed:

T = 40,000 exp[-0.33(37r + a1)1 5 1543 pounds,

T3 = 40,000 exp[-0.33(37r + a,) + (37r + 2a 1)] 5 52 pounds,

T4 = 40,000 expl-0.33[(3r + ial) + (37r + 2Q 1) + (3wr + 27rl)] } = 1.7 pounds,

T5 = 40,000 expl--0.33[(37r + a1 ) + 2(3r + a1 ) + 27r] 0.2 pound.

62

S- The location of forces and twisting moments are assumed as in figure 27A, and theresulting shear and moment stresses are shown in figure 27B. Using figure 27B, the followingmaximum values for shear force, twisting moment, and bending moment are obtained:

Fmax 38,507 pounds,

"Mmax =154,180 inch-pounds,

Mt,max = 245,484 inch-pounds.

Since the allowable loading limits are

Fa = l01,000 pounds,

Ma 1,135,000 inch-pounds,

"Mt,a =1,169,000 inch-pounds,

no loading limit has been exceeded.

Comparison of Examples

The three rope-wrapping configuration. are compared in table 13.

A Table 13. Comparison of maximum loads as fractions of allowed loads.

Maximum Load as Fraction of Allowed LimitExamples Fmax/Fa Mmax/Ma Mt,max/Mt,a

1 0.34 0.09 0.20

2 0.38 0.14 0.21

3 0.38 0.14 0.21

Based on these data, it can be concluded that, since maximum loading does not equal 50 per-

SL cent of the allowed load limit in any configuration, all are equally acceptable. However, it

should be remembered that smaller loads were applied to the bitt barrel in example 1 andthat there are nonstructural factors, e.g., ease of line handling, which can make a particularconfiguration more advantageous than others for a specific application.

63

barrel 2

~~ I F2 52- °.7t5'Mt -321

h=7in i

Mt 1 =245,163 F1 =4n,000I 1543 =38,4574 in

FR = 38,507 MMt, R

Part A. Forces and twsiting moments.

h, in

7 Mt =321 F =50

,:hear

4 ~ Mt 245,484 F 38,507

twisting moment M 148

bending 15,8moment max

"Mt, max = 245,484 Fmax =38,507• ., , ,p / ma..x ...245,484

b2ittO// 10o"/ / " base - 'xlo•,M 10

Part B. Shear and moment stress values.

Figure 27. Forces and moments for nonfigure-eight, reversing.wrap line•-" :- [configuration around two-barrel bitt in towing.

. ... '6

• - 6 4

I.

I,

K DISCUSSION

The following observations concerning the test results and performance of the testapparatus can be made.

1. Test data for consecutivw runs on the same rope and capstan combination show aconsiderable range of variation. There are two reasons for this:

Aj 1§ The textures of the ropes vary along their length because of nonuniformity ofmaterial and construction.

The rope creeps from side to side over the width of the capstan as new sections come

into contact with the test capstan. Surface wear and therefore the surface roughnessof the test capstan vary considerably from the marginal regions to the middle section"of the capstan surface, resulting in a large variation in the friction coefficient data.From a practical viewpoint, however, creeping is beneficial, since it provides an aver-age which represents both nonuniformities in the rope and irregularities on thecapstan surface.

2. More than half of the rope-capstan combinations tested indicated higher frictioncoefficients for a wet condition, i.e., wetting generally enhances both static and sliding fiic-tion. This factor is important for marine applications and should be further investigated.

3. The friction coefficients calculated from the low-load tests (table 5) when corn-pared to those obtained from high-load tests (table 6) show that the dry sliding frictioni coefficients between ropes A, B, E, F, G, and H and the standard capstan are not appreciably

affected by the rope's tension.

- - 4. Although the two capstans with stainless steel and NiCr/Cr3 C2 plasmalloy sur-faces provide exceptional life as far as the capstan surface is concerned, the outside fibers ofthe rope are trn off under sliding conditions. Consequently, the sliding friction is not a

- - true friction, but rather the resistance of the rope to tearing and mechanical failure of itsfibers. This chafing effect can seriously affect the life of the line.

5. In testing various ropes and capstans, under no conditions was chattering orvibration of the capstan-rope assembly observed. However, the situation under high pulsaL-ing tension may be different.

6. The results of bitt tests (table 7) agree with results for the capstan tests (table 6).

7. There is a large difference between the friction coefficient for bitt tests (table 7)and low-load capstan tests (table 5), The main reason lies in the different surfaces: thecapstans have a distinctly coarser surface than the pipes used to simulate the bitt barrels.

8. The test apparatus operated as expected in all three configurations (figures 6, 12,and 13). However, reduction of friction in the moving and sliding parts would definitely im-prove the accuracy of the results and reduce measurement errors.

4 9. Training of operators had to be repeated several times because of changes in avail-able personnel and proved to be a rather time-consuming task.

g• 65

REFERENCES

1. General Technology Company, Interim Report on Friction Coefficient of SyntheticRopes on Capstans, Bitts, Etc, prepared for Naval Undersea Center under contractNo. N66001-76-M-A814, September 1975.

2. Bureau of Ships, Drawing Number 805-921985, "Mooring Bitts, Two-Barrel, WeldedType."

3. Bureau of Ships, Drawing Number 805-1361959, "Mooring Bitts, Three-Barrel, WeldedType."

4. Roark, R. J., and Young, W. C., "Formulas f-'r Stress and Strain," Fifth Edition,McGraw-Hill Book Co.

5. Mark's Handbook, Seventh Edition, McGraw-Hill Book Company.

1 66